Convex conjugates of integral functionals

Abstract

We consider a convex integral functional on a functional space V andcompute its greatest extension to the algebraic bidual space V**, among all convex functions which are lower semicontinuous with respect tothe *-weak topology o(V** ; V*).Such computations are usually performed to extend these functionals to sometopological closures. In the present paper, no a priori topological restrictionsare imposed on the extended domain. As a consequence, this extended functionalis a valuable first step for the computation of the exact shape of the minimizersof the conjugate convex integral functional subject to a convex constraint,in full generality: without constraint qualification. These convex integralfunctionals are sometimes called entropies, divergences or energies. Our proofsmainly rely on basic convex duality and duality in Orlicz spaces.